Gottfried
Helms

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Numbertheoretical matrices

In this
page I'm presenting a collection of basic numbertheoretical matrices, for instance
the Pascalmatrix (binomialmatrix), matrices of Bernoulli-/Euler- and other polynomials,
Stirling-numbers and some interesting and systematic relations between them. (The
matrices are lower-triangular number-arrays of infinite dimension.)

This
collection is primarily a compilation of that matrices and their basic
properties. Then their relations are presented and discussed. The occuring
identities are in most cases well known, but especially the
eigensystem-decompositions (and their consequences) seem to be unique in this
collection.

Most of
the relations were found heuristically by playing around with that matrices
in the style of experimenting with a toolbox. The proofs for the given
identities and relations shall be added as I have time (and as far as I'm
able to derive them). When I was looking in literature (mostly internet
resources) I found most of these identities spreaded over various resources:
here I try to put them together in a more systematic context.

An
unexpected but interesting extension popped up very soon: the problems of divergent
summation: matrix-operations with that (in most cases) integer-numberarrays
means to assign values to divergent summation in a very natural way.

Also the
references section is far from being complete. I'll add the references while
I'm extending the already present chapters as well while I'm creating the
missing ones.

Thanks to
the developers of the program Pari/GP, with which I computed the examples,
and which they offered to the math-community for free. The matrix-bitmaps are
created with my own user-interface Pari-TTY, which the interested reader may
also download for free.

Gottfried Helms
12'2006 - 06'2008

Intro and notation

The basic
definitions and conventions, the short names of the basic matrices.

Stirlingnumbers : the matrices St1 and St2 of
Stirlingnumbers of 1st and 2nd kind, Bell-numbers. (11.02.07)

2 More matrix-relations
of interest

The Vandermondematrix ZV seems more and more the core matrix of all that considerations. This
is not surprising: it is vertically read the matrix of zeta-sequences and
horizontally read of the geometric series: the most basic aspects that may be
studied in number-theories.

3 Special matrices and
interesting details

The
"Gaussian"-matrix occured to me when I tried to find derivatives
and integrals of the gaussian function. The matrix is then the triangle of
coefficients of z which are needed to express the derivatives. It occured,
that even this matrix is closely related to the binomialmatrix by the same
subdiagonal/matrix-exponential-scheme as a very nice and simple
generalization.

Generalized Bernoulli-Recursion: a surprising recursion formula
to define the Bernoulli-numbers. This formula can be parametrized, and very
natural selections of parameters lead to a whole family of numbertheoretical
sequences, which shows a basic simple relationship between harmonic,
geometric, constant, reciprocal binomial series together with the
bernoulli-numbers and "eta"-numbers. (14.02.07)

Sums of like powers: an approach to find an
equivalent to bernoulli-polynomials by means of matrix-algebra. The
Faulhaber/Bernoulli-matrix is recovered by elementary means and the
zeta-connection re-established (20.5.2007).

4 divergent Summation
using matrices

Summation:Introductory remarks about
divergent summation: Cesaro/Euler-summation in matrix-representation; A
direct summation-approach for some special applications

A formula
for infinite "geometric series" of Power-towers is presented.

The
formula seems to allow to extend the domain, as it is known for infinite powertowers,
to values s>e1/e (4.7.2007)

Older versions:

Pascalmatrix : introduces the concept, the Pascalmatrix,
Gp-matrix as eigensystem, St2/St1-matrices; can be used until the chapter
"Bernoulli-/GP-matrix" is rewritten. (German version)

Summation:
divergent summation using the binomial and other matrices / still manuscript
version. Touches the most aspects which are discussed in the newer articles
in terms of heuristic findings and propositions.